The Number of Impedances of an n Terminal Network

L I N E A R passive network having n accessible terminals may be completely represented by an equivalent direct impedance network, 1 consisting of branches, devoid of mutual impedance, connecting the terminals in pairs. The number of elements (branches) in this representation is equal to the number of combinations of n things taken two at a time, i.e., n(n -- 1). Each of the elements is defined by an impedance measured by energizing between one of the terminals it connects and the remaining terminals connected together and taking the ratio of the driving voltage to the current into the other terminal it connects. The network then is represented by a particular set, of n(n -- 1) members, of impedances measurable at its terminals; as will appear later, the set is of short-circuit transfer impedances. The direct impedance network is one among many network representations; it is taken as illustrative of two aspects, (i) the necessity of a certain number of elements n(n -- 1) and (ii) the expression of these elements in terms of measurable impedances. It is well known that any linearly independent set, of n(n -- 1) members, of the measurable impedances of an w-terminal network will serve as a network representation; hence the enumeration of representations may be taken in two steps, the first of which, the enumeration of measurable impedances, is dealt with in the present paper.